The Service Rate Region (SRR) serves as a critical metric for evaluating the concurrent service capacity of distributed storage systems. While several works have characterized the SRR for MDS codes and first order Reed-Muller codes, for high-order Reed-Muller codes the problem becomes way more complicated and only partial results were given by Ly, Soljanin, and Lalitha [IEEE ISIT 2025]. In this paper, we refine the SRR analysis by explicitly characterizing the intersection patterns of recovery sets for high-order Reed-Muller codes, deriving the exact region for the case m=r+1 and providing new types of strictly tighter constraints to bridge the gap between existing approximations and the exact SRR polytope.

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